CZ-SK-MO 2014 Problem 5

Given is the acute triangle $ABC$ . Let us denote $k$ a circle with diameter $AB$ . Another circle, tangent to $AB$ at point $A$ and passing through point $C$ intersects the circle $k$ at point $P, P \ne A$ . Another circle which touches AB at point $B$ and passes point $C$ , intersects the circle $k$ at point $Q, Q \ne B$ . Prove that the intersection of the line $AQ$ and $BP$ lies on one of the sides of angle $ACB$ .

(Peter Novotný)